| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s + 8·8-s + 9·9-s + 10·10-s − 19·11-s − 12·12-s − 33·13-s − 15·15-s + 16·16-s − 64·17-s + 18·18-s + 141·19-s + 20·20-s − 38·22-s − 51·23-s − 24·24-s + 25·25-s − 66·26-s − 27·27-s + 216·29-s − 30·30-s − 290·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.520·11-s − 0.288·12-s − 0.704·13-s − 0.258·15-s + 1/4·16-s − 0.913·17-s + 0.235·18-s + 1.70·19-s + 0.223·20-s − 0.368·22-s − 0.462·23-s − 0.204·24-s + 1/5·25-s − 0.497·26-s − 0.192·27-s + 1.38·29-s − 0.182·30-s − 1.68·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 19 T + p^{3} T^{2} \) |
| 13 | \( 1 + 33 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 - 141 T + p^{3} T^{2} \) |
| 23 | \( 1 + 51 T + p^{3} T^{2} \) |
| 29 | \( 1 - 216 T + p^{3} T^{2} \) |
| 31 | \( 1 + 290 T + p^{3} T^{2} \) |
| 37 | \( 1 + 109 T + p^{3} T^{2} \) |
| 41 | \( 1 + 457 T + p^{3} T^{2} \) |
| 43 | \( 1 - 184 T + p^{3} T^{2} \) |
| 47 | \( 1 + 313 T + p^{3} T^{2} \) |
| 53 | \( 1 + 319 T + p^{3} T^{2} \) |
| 59 | \( 1 + 44 T + p^{3} T^{2} \) |
| 61 | \( 1 - 368 T + p^{3} T^{2} \) |
| 67 | \( 1 - 216 T + p^{3} T^{2} \) |
| 71 | \( 1 + 314 T + p^{3} T^{2} \) |
| 73 | \( 1 + 602 T + p^{3} T^{2} \) |
| 79 | \( 1 - 112 T + p^{3} T^{2} \) |
| 83 | \( 1 + 712 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1018 T + p^{3} T^{2} \) |
| 97 | \( 1 + 584 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.774080577651112827724403856949, −7.64689561999095164678119135008, −6.97492665084509286783018563433, −6.15794890883763502583910430997, −5.22131330145937238200998406720, −4.85493653009622739768585350098, −3.59450177840308737835731572062, −2.57598947397558541772138698019, −1.49653605426958114493278083437, 0,
1.49653605426958114493278083437, 2.57598947397558541772138698019, 3.59450177840308737835731572062, 4.85493653009622739768585350098, 5.22131330145937238200998406720, 6.15794890883763502583910430997, 6.97492665084509286783018563433, 7.64689561999095164678119135008, 8.774080577651112827724403856949
